On nn-plectic manifolds

Definition

For (X,ω)(X,\omega) an n-plectic manifold, a vector fieldv∈Γ(TX)v \in \Gamma(T X) is called a Hamiltonian vector field if is contraction with the (n+1)(n+1)-form ω\omega is exact: there is α∈Ωn−1(X)\alpha \in \Omega^{n-1}(X) such that

On nn-plectic smooth ∞\infty-groupoids

We discuss now the notion of Hamiltonian vector fields in the full generality internal to a cohesive (∞,1)-toposH\mathbf{H}. We write out the discussion for the case H=\mathbf{H} = Smooth∞Grpd for convenience, but any other choice of cohesive (∞,1)(\infty,1)-topos works as well.

we call the ∞\infty-Lie algebra of Hamiltonian vector fields on (X,ω)(X,\omega).

More explicitly:

Definition

A Hamiltonian diffeomorphismϕ\phi onon (X,ω)(X, \omega) is an element ϕ:X→≃X\phi \colon X \stackrel{\simeq}{\to} X in the automorphism ∞-groupϕ∈Aut(X)\phi \in \mathbf{Aut}(X) such that it fits into a diagram of the form

Proof

To compute the Lie algebra of this, we need to consider smooth 1-parameter families of Hamiltonian diffeomorphisms and differentiate them.

Assume first that the prequantum line bundle is trivial as a bundle, with the connection 1-form of ∇\nabla given by a globally defined A∈Ω1(X)A \in \Omega^1(X) with dA=ωd A = \omega. Then the existence of the diagram in def. 4 is equivalent to the condition

(ϕ(t)*A−A)=dα(t),
(\phi(t)^* A - A) = d \alpha(t)
\,,

where α(t)∈C∞(X)\alpha(t) \in C^\infty(X). Differentiating this at 0 yields the Lie derivative

ℒvA=dα′,
\mathcal{L}_v A = d \alpha'
\,,

where vv is the vector field of which t↦ϕ(t)t \mapsto \phi(t) is the flow and where α′:=ddtα\alpha' := \frac{d}{dt} \alpha.